研究目的
To derive the frozen Gaussian approximation for high-frequency wave propagation in periodic media and establish its convergence to the true solution, addressing the computational challenges of conventional methods.
研究成果
The frozen Gaussian approximation is derived and shown to converge to the true solution with an error of O(ε) in the L2 norm for high-frequency wave propagation in periodic media, providing an efficient alternative to costly conventional methods.
研究不足
The analysis assumes that the Bloch bands do not intersect and are separated, and the Hamiltonian is subquadratic. The method is limited to the semiclassical regime with small ε, and numerical implementation details are deferred to a companion paper.
1：Experimental Design and Method Selection:
The study uses asymptotic analysis and Bloch decomposition to formulate the frozen Gaussian approximation (FGA) for the semiclassical Schr?dinger equation with periodic potentials. Theoretical models include Hamiltonian flows and Fourier integral operators.
2：Sample Selection and Data Sources:
The analysis is theoretical, focusing on mathematical derivations without empirical data or specific samples.
3：List of Experimental Equipment and Materials:
No physical equipment or materials are used; the work is computational and analytical.
4：Experimental Procedures and Operational Workflow:
The methodology involves deriving the FGA formulation, establishing convergence results through rigorous mathematical proofs, and referencing numerical algorithms in a companion paper.
5：Data Analysis Methods:
The analysis relies on mathematical techniques such as Bloch decomposition, asymptotic expansions, and error estimates using norms like L2.
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